Tuesday, April 23, 2013

Holes are emergent quasi-particles

When I first taught Solid State Physics [following Ashcroft and Mermin] I would introduce holes as the absence of an electron. I then discuss the effective mass of "electrons" and holes near the bottom and top of bands, respectively. I would not introduce the concept of a quasi-particle until several weeks later when I discussed electron-electron interactions and Landau's Fermi liquid theory.

Now I do it differently.
I explain how holes are an example of a quasi-particle with a positive charge and an effective mass [which can be significantly larger or smaller than the free electron mass].
This is a nice "simple" example of emergence. When you put interacting particles ["non-interacting" electrons interacting with a nuclei in a periodic lattice] together new entities emerge which has properties that are qualitatively different from the constituent particles.
[Aside: it is interesting that the only "interactions" between the electrons themselves are those associated with Fermi statistics]

As an aside, I then try to get students to think about some of the philosophical questions asking them to vote on and discuss the following questions:

6 comments:

Yes, electrons are real in the same way quarks are real: they are fundamental particles of the universe.

Yes, holes are real in the same way chairs are real. They aren't fundamental particles of the universe, but collections of such particles and their interactions, with a particular set of properties that is distinct enough that it's useful to give them a name.

I also think it helps to give other examples of emergence, like how bubbles rise in fizzy drinks because gravity is pulling the drink down, not pushing the bubble up. I did this in a blog post on electrons and holes, though that's more aimed at the public rather than students in a solid state class, who should presumably know a thing or two about emergent properties.

A quantum optics researcher once told me that for him a photon is a click in the detector, which makes sense I guess. Going by which, I would imagine that an indescribable (for theoreticians) or its superset of an unobservable (for experimentalists) object might qualify as unreal.

I think, the doubt about 'reality' can be rephrased as: Can we detect them?

Like Majorana fermions should exist, in theory, at the ends of a 1D superconductors. But if we cannot detect them, we cannot confirm its real existence.

We can go back to Dirac's hole theory which were not attainable due to filled Dirac sea. After Carl Anderson's discovery we know they are basically positrons.

Similarly back to the condensed matter, graphene came up showing relativistic Dirac electrons. And again many properties of Dirac electrons in graphene have been tested (eg. Klein tunneling) in the lab.

I think for quasiparticles, ARPES and STM measurements are enough to talk about their existence in the real world. Even though they are basically electrons that got "dressed" due to interactions.

The picture I find most compelling, is that electrons exist in condensed matter, because it simplifies the math. And if electrons exist, then holes really are quasiparticles. If our choice of description were that holes exist, then electrons would be the quasiparticle. Wouldn't they?

However electrons are excitations themselves. To a particle physicist, this is an excitation of a field, because that's the way our picture, because of the math, has evolved.

But isn't the best we can say is that these are all effective theories, relevant at the scales the person is interested in? And even more confusing, each effective theory is one choice of many possible effective theories.

For instance, I can take interacting electrons, and rewrite my action in terms of non-interacting electrons, which are coupled to bosonic fields. Because the transformation is exact, it's just an 'accident' that I started with interacting electrons. So on that level, I can't really claim that one of the other is what's "really" happening.

Similarly, I published something on the \nu=5/2 fractional quantum Hall effect last year. I had to decide: do I use composite fermions, or composite bosons to do this calculation? They both give good results! But then, maybe "bare" electrons are composite fermions with more flux tubes attached. The transformation works both ways.

And can't a Chern Simon's (like?) theory be used to map fermions onto hard-core bosons with special properties?

But then, as is pointed out above in different ways, I could write down a Green's function, which has poles, and then I could say 'aha', that's what's real!

So I think I agree with some of the statements above, that the poles of correlation functions, when they agree with experiment, are the most we can say about "reality". And the fields we use to write down the correlation function, we can call particles, because they "work".

But I don't believe we can make statements about "fundamental particles" or anything like this. I think we write down a theory which mimics the math-adherence of the physical world, and then calculate correlation functions and if there's a pole, we define a particle. If the correlation uses a bare Green's function, we tend to call it a particle, and if it's something else, we start to use words like quasiparticles. But then, this is probably just a function of our choice of initial field.

So I think electrons and holes are both really quasiparticles, but one of many possible descriptions.

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About Me

I have fun at work trying to use quantum many-body theory to understand electronic properties of complex materials.
I am married to the lovely Robin and have two adult children and a dog, Priya (in the photo). I also write an even more personal blog Soli Deo Gloria [thoughts on theology, science, and culture]

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